GRÖBNER THEORY AND TROPICAL GEOMETRY ON SPHERICAL VARIETIES

Let G be a connected reductive algebraic group. We develop a Gröbner theory for multiplicity-free G-algebras, as well as a tropical geometry for subschemes in a spherical homogeneous space G/H. We define the notion of a spherical tropical variety and prove a fundamental theorem of tropical geometry in this context. We also propose a definition for a spherical amoeba in G/H using Cartan decomposition. Our work partly builds on the previous work of Vogiannou on spherical tropicalization and in some ways is complementary.